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Mirrors > Home > MPE Home > Th. List > hlcomd | Structured version Visualization version GIF version |
Description: The half-line relation commutes. Theorem 6.6 of [Schwabhauser] p. 44. (Contributed by Thierry Arnoux, 21-Feb-2020.) |
Ref | Expression |
---|---|
ishlg.p | β’ π = (BaseβπΊ) |
ishlg.i | β’ πΌ = (ItvβπΊ) |
ishlg.k | β’ πΎ = (hlGβπΊ) |
ishlg.a | β’ (π β π΄ β π) |
ishlg.b | β’ (π β π΅ β π) |
ishlg.c | β’ (π β πΆ β π) |
ishlg.g | β’ (π β πΊ β π) |
hlcomd.1 | β’ (π β π΄(πΎβπΆ)π΅) |
Ref | Expression |
---|---|
hlcomd | β’ (π β π΅(πΎβπΆ)π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlcomd.1 | . 2 β’ (π β π΄(πΎβπΆ)π΅) | |
2 | ishlg.p | . . 3 β’ π = (BaseβπΊ) | |
3 | ishlg.i | . . 3 β’ πΌ = (ItvβπΊ) | |
4 | ishlg.k | . . 3 β’ πΎ = (hlGβπΊ) | |
5 | ishlg.a | . . 3 β’ (π β π΄ β π) | |
6 | ishlg.b | . . 3 β’ (π β π΅ β π) | |
7 | ishlg.c | . . 3 β’ (π β πΆ β π) | |
8 | ishlg.g | . . 3 β’ (π β πΊ β π) | |
9 | 2, 3, 4, 5, 6, 7, 8 | hlcomb 28394 | . 2 β’ (π β (π΄(πΎβπΆ)π΅ β π΅(πΎβπΆ)π΄)) |
10 | 1, 9 | mpbid 231 | 1 β’ (π β π΅(πΎβπΆ)π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1534 β wcel 2099 class class class wbr 5142 βcfv 6542 Basecbs 17171 Itvcitv 28224 hlGchlg 28391 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-hlg 28392 |
This theorem is referenced by: hlcgreulem 28408 opphllem4 28541 opphllem5 28542 opphl 28545 hlpasch 28547 lnopp2hpgb 28554 colhp 28561 cgrahl1 28607 cgrahl2 28608 cgrahl 28618 cgracol 28619 dfcgra2 28621 sacgr 28622 acopy 28624 acopyeu 28625 inaghl 28636 tgasa1 28649 |
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